| 1. |
- Freij, Ragnar, 1984, et al.
(författare)
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Partially ordered secretaries
- 2010
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 15, s. 504-507
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Tidskriftsartikel (refereegranskat)abstract
- The elements of a finite nonempty partially ordered set are exposed at independent uniform times in [0, 1] to a selector who, at any given time, can see the structure of the induced partial order on the exposed elements. The selector’s task is to choose online a maximal element. This generalizes the classical linear order secretary problem, for which it is known that the selector can succeed with probability 1=e and that this is best possible. We describe a strategy for the general problem that achieves success probability at least 1=e for an arbitrary partial order.
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| 2. |
- Hessler, Martin, et al.
(författare)
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Edge cover and polymatroid flow problems
- 2010
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Ingår i: Electronic Journal of Probability. - : Institute of Mathematical Statistics. - 1083-6489. ; 15, s. 2200-2219
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Tidskriftsartikel (refereegranskat)abstract
- In an n by n complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This so-called minimum edge cover problem is a relaxation of perfect matching. We show that the large n limit cost of the minimum edge cover is W(1)(2) + 2W(1) approximate to 1.456, where W is the Lambert W-function. In particular this means that the minimum edge cover is essentially cheaper than the minimum perfect matching, whose limit cost is pi(2)/6 approximate to 1.645. We obtain this result through a generalization of the perfect matching problem to a setting where we impose a (poly-)matroid structure on the two vertex-sets of the graph, and ask for an edge set of prescribed size connecting independent sets.
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| 3. |
- Wästlund, Johan, 1971
(författare)
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The mean field traveling salesman and related problems
- 2010
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Ingår i: Acta Mathematica. - : International Press of Boston. - 0001-5962. ; 204:1, s. 91-150
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Tidskriftsartikel (refereegranskat)abstract
- The edges of a complete graph on n vertices are assigned i.i.d. random costs from a distribution for which the interval [0, t] has probability asymptotic to t as t -> 0 through positive values. In this so called pseudo-dimension 1 mean field model, we study several optimization problems, of which the traveling salesman is the best known. We prove that, as n -> a, the cost of the minimum traveling salesman tour converges in probability to a certain number, approximately 2.0415, which is characterized analytically.
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